The masses and radii of earth and moon are $M_{1}, R_{1} and M_{2}, R_{2}$ respectively. Their centers are distance $d$ apart. The minimum velocity with which a particle of mass $m$ should be projected from a point midway between their centre so that it escapes to infinity is:-

2 \sqrt{\frac{G}{d} (M_{1} + M_{2})}

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2 \sqrt{\frac{2 G}{d} (M_{1} + M_{2})}

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2 \sqrt{\frac{G m}{d} (M_{1} + M_{2})}

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2 \sqrt{\frac{G m (M_{1} + M_{2})}{d (R_{1} + R_{2})}}

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Solution

The correct option is A $2 \sqrt{\frac{G}{d} (M_{1} + M_{2})}$
Gravitational potential energy at midway between their centre,
$P E = \frac{- G M_{1} m}{\frac{d}{2}} - \frac{G M_{2} m}{\frac{d}{2}} = - \frac{2 G m}{d} (M_{1} + M_{2})$
For projection of particle from mid point to infinity, Total energy must be equal to more than zero.
$\Rightarrow K E = | \begin{matrix} P E \end{matrix} |$

$\Rightarrow \frac{1}{2} m v^{2} = \frac{2 G m}{d} (M_{1} + M_{2}) \Rightarrow v = 2 \sqrt{\frac{G (M_{1} + M_{2})}{d}}$

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The masses and radii of the earth and moon are $M_{1}, R_{1} a n d M_{2}, R_{2}$ respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is